CLASSICAL MOTIVATION FOR THE RIEMANN–HILBERTCORRESPONDENCE

BRIAN CONRAD

These notes explain the equivalence between certain topological and coherentdata on complex-analytic manifolds, and also discuss the phenomenon of “regularsingularities” connections in the 1-dimensional case. It is assumed that the reader isfamiliar with the equivalence of categories between the category of locally constantsheaves of sets on a topological space X and the category of covering spaces over X,and also (for a pointed space (X, x0)) with the resulting equivalence of categoriesbetween the category of locally constant sheaves of sets on X and left π1(X, x0)-setswhen X is connected and “nice” in the sense that it has a base of opens that arepath-connected and simply connected. (For example, it is a hard theorem that anycomplex-analytic space is “nice”). These equivalences will only be used in the caseof complex manifolds, but for purposes of conceptual clarity we give some initialconstruction without smoothness restrictions.

Let us first set some conventions. As usual in mathematics, we fix an algebraicclosure C of R, endowed with its unique absolute value extending that on R. Weshall work with arbitrary complex-analytic spaces (the analytic counterpart to C-schemes that are locally of finite type), but the reader who is unfamiliar with thistheory will not lose anything (nor run into problems in the proofs) by requiringall analytic spaces to be manifolds (in which case what we are calling a “smoothmap” is a submersion by another name). For technical purposes it is importantthat the theory of D-modules works without smoothness restrictions. Hence, forthe analytically-inclined reader who wishes to avoid smoothness restrictions wenote that the global method of construction of the relative de Rham complex Ω•X/S

for any morphism of schemes f : X → S as in [5, IV4, §16.6] works verbatim inthe complex-analytic case and is compatible with the analytification functor fromlocally finite type C-schemes to complex-analytic spaces.

Notation and terminology. If (X, OX) is a locally ringed space and F isan OX -module then F∨ denotes H omOX

(F ,OX) and F (x) denotes Fx/mxFx;we call F (x) the fiber at x (to be distinguished from the stalk Fx at x). We willfreely pass between the categories of vector bundles and locally free sheaves, andso if V is a vector bundle then we may write V (x) to denote its fiber over a pointx in the base space. If Σ is a set and X is a topological space then Σ denotes theassociated constant sheaf of sets on X.

The analytic space SpC is SpecC considered as a 0-dimensional complex mani-fold. Also, Z(1) denotes the kernel of exp : C → C×; this is a free Z-module of rank1 but it does not have a canonical basis. Finally, a local system (of sets, abeliangroups, etc.) on a topological space is a locally constant sheaf (of sets, abeliangroups, etc.) on the space.

1

2 BRIAN CONRAD

1. Basic definitions and constructions

Let π : X → Y be a map of analytic spaces, and E a coherent sheaf on X. Aconnection on E relative to Y is a map of abelian sheaves

∇ : E → Ω1X/Y ⊗OX

E

that satisfies the Leibnitz rule

∇(fs) = dX/Y f ⊗ s + f · ∇(s)

for local sections s of E and f of OX ; in particular, ∇ is OY -linear. Such pairs(E ,∇) form a category in an evident manner. The set of connections on E relativeto Y is a principal homogeneous space under HomX(E ,Ω1

X/Y ⊗E ) because ∇−∇′is OX -linear for any two connections on E relative to Y .

Example 1.1. A basic example with Y = SpC is E = OX ⊗C Λ for a local systemΛ of finite-dimensional C-vector spaces, taking ∇ = dX⊗1. This connection clearlykills Λ, and it is the unique connection on E with this property (due to the Leibnitzrule). If X is a manifold then ker dX = C, and so ker∇ = Λ in the smooth case.

More generally, for any Y we may consider E = OX ⊗π−1OYΛ for a locally free

π−1OY -module Λ with finite rank. In this case, ∇ = dX/Y ⊗1 is a connection on Ethat kills the image of Λ in E and is uniquely determined by this property. WhenX is Y -smooth then ker∇ = Λ because ker dX/Y = π−1OY and Λ is locally freeover π−1OY .

Example 1.2. Assume that X is Y -smooth and E is a vector bundle. Choose anopen U ⊆ X such that E |U admits a basis ej and U ' Y × V for an open V inCn. We have unique expansions

∇(ej) =∑i,k

Γkijdzi ⊗ ek,

where zi is the standard coordinate system on Cn and the Γkij ’s are sections of

OX over U . For a general local section s =∑

sjej of E |U ,

∇(s) =∑i,k

∂sk

∂zi+∑

j

Γkijsj

dzi ⊗ ek.

Thus, ∇(s) = 0 is a system of first-order homogeneous linear PDE’s. Conversely,given sections Γk

ij of OX over U we may use this formula to define a connection onU over Y . The Γk

ij ’s are the Christoffel symbols of the connection relative to thelocal choices of coordinates and basis. In the special case when X is Y -smooth withpure relative dimension 1, we write Γij rather than Γ1

ij , and (Γij) is the connectionmatrix with respect to the chosen basis of E |U and the choice of relative coordinatez1.

A connection is to be viewed as a mechanism for differentiating sections alongvector fields: if ~v is a relative vector field on X over Y (i.e., an OY -linear derivationof OX), then the composite ∇~v = (~v ⊗ 1) ∇ : E → E satisfies

∇~v(fs) = ~v(f) · s + f · ∇~v(s).

The operation ∇~v is “differentiation along the vector field ~v.” Note that ~v 7→ ∇~v

is OX -linear.

4 BRIAN CONRAD

If E is a vector bundle on X with a connection ∇ relative to Y , then the dualconnection ∇∨ on the dual E ∨ is defined via

∇∨(`) = dX/Y `− (1⊗ `) ∇ ∈ HomOU(E |U ,Ω1

X/Y |U ) = Γ(U,Ω1X/Y ⊗OX

E ∨)

for ` ∈ E ∨(U); the motivation for this is to ensure that the identity

dX/Y (`(s)) = (∇∨(`))(s) + (1⊗ `)(∇(s))

holds for all local sections s of E |U . The assignment ` 7→ ∇∨(`) satisfies the Leibnitzrule, so ∇∨ is indeed a connection on E ∨ relative to Y . It is not difficult to verifythat ∇∨∨ = ∇ via the isomorphism E ∨∨ ' E .

Example 1.4. For vector bundles E1 and E2 on X equipped with connections ∇1

and ∇2 relative to Y , the isomorphism H omX(E1,E2) ' E2 ⊗ E ∨1 allows us todefine a connection ∇ = ∇2⊗∇∨1 on this Hom-sheaf. Calculations with elementarytensors show that for a section ϕ : E1|U → E2|U over an open U ,

∇(ϕ) ∈ Γ(U,H omX(E1,E2 ⊗ Ω1X/Y )) = HomU (E1|U ,E2 ⊗ Ω1

X/Y |U )

is given by ∇2 ϕ − (1 ⊗ ϕ) ∇1. In particular, ∇(ϕ) = 0 if and only if ϕ iscompatible with the connections ∇1 and ∇2, and for (E2,∇2) = (OX ,dX/Y ) thisrecovers the dual connection.

Finally, for any commutative square

(1.1) X ′ h′ //

π′

X

π

Y ′

h// Y

and (E ,∇) on X relative to Y , the pullback connection ∇′ on X ′ (relative to Y ′) isdefined as follows. Let η : h′

∗Ω1X/Y → Ω1

X′/Y ′ be the canonical map. The map

∇′ : h′∗E = OX′ ⊗h′−1OX

h′−1

E → Ω1X′/Y ′ ⊗OX′ h′

∗E

defined by

∇′(f ′ ⊗ h′−1

s) = df ′ ⊗ h′∗(s) + f ′ · (η ⊗ 1)(h′∗(∇(s)))

is readily checked to be well-defined (i.e., for any local sections f ′, f , and s, thepairs (f ′ · h′−1(f), h′−1(s)) and (f ′, h′−1(f) · h′−1(s)) are sent to the same place),and it satisfies the Leibnitz rule.

The pullback connection is often denoted h′∗(∇), though it also depends on the

left and bottom sides of (1.1), and it is uniquely characterized by the propertythat it sends h′

−1(s) to (η ⊗ 1)(h′∗(∇(s))). This characterization implies thatthe pullback operation on connections is naturally transitive and associative withrespect to composite pullbacks. Local calculations also show that formation of dualand tensor-product connections is compatible with pullback.

Example 1.5. The construction in Example 1.1 is compatible with duality, tensorproduct, and pullback. To be precise, let Λ and Λ′ be locally free π−1OY -modulesof finite rank, and let E = OX ⊗π−1OY

Λ and E ′ = OX ⊗π−1OYΛ′. Let ∇Λ and

∇Λ′ be the unique connections on E and E ′ whose kernel contains Λ and Λ′. Withrespect to the isomorphisms

E ∨ ' OX ⊗π−1OYΛ∨, E ⊗OX

E ′ ' OX ⊗π−1OY(Λ⊗π−1OY

Λ′),

CLASSICAL MOTIVATION FOR THE RIEMANN–HILBERT CORRESPONDENCE 5

a local calculation shows that the connections ∇∨Λ and ∇Λ ⊗∇Λ′ kill the images ofΛ∨ and Λ⊗ Λ′, and so uniqueness implies ∇∨Λ = ∇Λ∨ and ∇Λ ⊗∇Λ′ = ∇Λ⊗Λ′ . Ina similar manner, for any commutative square as in (1.1), the pullback connectionh′∗(∇Λ) is identified with ∇h∗Λ.

There is a fundamental link between local systems and vector bundles withconnection. The starting point is the following reformulation of the local uniquenessof solutions to first-order ODE’s with an initial condition:

Lemma 1.6. If X is smooth and (E ,∇) is a vector bundle on X with a connection,then for the C-module ker∇ on X the C-linear map (ker∇)x → E (x) is injectivefor all x ∈ X. In particular, all stalks of ker∇ are finite-dimensional over C.

The sets X≤d = x ∈ X | dim(ker∇)x ≤ d are a locally finite collection ofclosed sets in X, and ker∇ has locally constant restriction to X≤d −X≤(d−1) forall d. In particular, ker∇ is locally constant on X if and only if x 7→ dim(ker∇)x

is a locally constant function on X.

In §2 we shall determine when (ker∇)x = E (x) for all x ∈ X (and so in particularker∇ is locally constant).

Proof. We may work locally on X, so we can assume X = Bn is a polydisc and Eis globally free. Let z1, . . . , zn be the standard coordinates on Bn, and let ej bea basis of E . The sections dzi ⊗ ek give a basis of Ω1

X ⊗ E , and so if we expand∇(ej) with respect to this basis we see that the general condition ∇(

∑siei) = 0 is

a system of first-order linear PDE’s in the si’s with coefficients expressed in termsof the Γk

ij ’s as in Example 1.2. The uniqueness theorem for ODE’s with an initialcondition may be applied to restrictions along analytic curves to conclude that asection of ker∇ over a connected open U is uniquely determined by the specificationof its value at a single point x0. Thus, (ker∇)x → E (x) is indeed injective for allx and the sets X −X≤d are open (so each X≤d is closed). It is likewise clear thatthe collection of closed sets X≤dd is locally finite on X.

Pick x ∈ X≤d−X≤(d−1), so d = dim(ker∇)x. By shrinking X around x we mayarrange that there is a d-dimensional subspace V ⊆ (ker∇)(X) with V mappingisomorphically onto (ker∇)x. It follows that OX⊗C V → E is a map of vector bun-dles on the manifold X such that it induces an injection on x-fibers. By shrinkingX around x, we may therefore arrange that this map is a subbundle, and hencethe induced map V = V x′ → E (x′) is injective for all x′ ∈ X. This factors throughthe subspace (ker∇)x′ , so the map φ : V → ker∇ induces an injection on stalksat all points. Thus, φ is a subsheaf inclusion. The pullback of φ over the subsetX≤d−X≤(d−1) is therefore an isomorphism on stalks and hence is an isomorphism.Thus, ker∇ indeed has locally constant restriction to X≤d −X≤(d−1).

Let us consider a fundamental example that illustrates the general problems tobe considered in what follows. Let D∗ = 0 < |q| < 1 and let f : E → D∗ bethe analytic family of elliptic curves with q-fiber C×/qZ; to be rigorous, E is thequotient of C× ×D∗ by the action of the D∗-group Z ×D× with (n, q) acting by(t, q) 7→ (qnt, q). The dual sheaf Λ = (R1f∗C)∨ is a local system on D∗ whose fiberat q is naturally identified with H1(Eq,C) because f is a fibration.

Let e1(q0) ∈ H1(Eq0 ,C) be the pushforward of (1/2πi)σi ∈ H1(C×,C), whereσi is the path in C× that goes once around the origin in the i-counterclockwisedirection; note that e1(q0) is independent of i. Let e2(q0) ∈ H1(Eq0 ,C) be the class

6 BRIAN CONRAD

of the closed loop obtained by pushing forward a choice of path in C× joining 1 toq0; this is only canonical modulo Ce1(q0). Let γi be an i-counterclockwise generatorof π1(D∗, q0). For a loop γ representing a homotopy class g in π1(D∗, q0) and anye ∈ H1(Eq0 ,C) = Λq0 , if we transport the homology class e along γ from γ(0) = q0

to γ(1) = q0 via local constancy of Λ (or better: via the unique identification γ∗Λ 'Λq0 over [0, 1] extending the canonical isomorphism (γ∗Λ)0 ' Λq0 on 0-fibers), thenwe obtain g(e). Thus, γi(e1(q0)) = e1(q0) and γi(e2(q0)) is the concatenation of thepaths e2(q0) and 2πie1(q0), so γi(e2(q0)) = e2(q0) + 2πie1(q0).

The non-triviality of the π1-action implies that Λ does not extend to a localsystem on D. However, and this is the key point, the vector bundle E = OD∗ ⊗C Λdoes naturally extend to a rank-2 vector bundle on D. The reason this happensis that there exist differential equations on D∗ whose local solutions in O exhibitthe same monodromy behavior as e2 when we analytically continue them along γi,and so a suitable O-linear combination kills the non-trivial monodromy from Λ.More precisely, consider a local solution log to the differential equation df = dq/q.If we move around the circle through q0 in the i-counterclockwise direction thenwhen we return to q0 we have the local solution log +2πi. Thus, the local sectionlog⊗e1 − e2 is invariant under analytic continuation along a generating loop of π1,and so it extends to a global section v2 of E that is independent of i but is onlycanonical modulo Cv1. The global sections v1 = e1 and v2 give a basis of E overOD∗ , but the Cv1-ambiguity in the construction of v2 implies that the intrinsicstructure on E is a short exact sequence

0 → OD∗ → E → OD∗ → 0

with the subobject having basis v1 and the quotient having basis v2 mod Cv1. Thechoice of v2 splits this sequence.

With respect to the OD∗ -basis v1, v2, the connection ∇ = d ⊗ 1 on E withkernel Λ is given by

av1 + bv2 7→dq

q⊗ ((qa′ + b)v1 + qb′v2)

for local sections a and b of OD∗ (with derivatives a′ and b′), and so when we usethe vj ’s to extend E to a free vector bundle E ′ = O⊕2

D we see that ∇ extends to amap ∇′ : E ′ → Ω1

D(0)⊗ODE ′ that is a connection with a simple pole at the origin.

Changing v2 modulo Cv1 gives an isomorphic structure of the same type over D.The construction of (E ′,∇′) may appear to be ad hoc, but we shall prove in §4 thatthe simple-pole property uniquely characterizes it up to unique isomorphism as anextension of (E ,∇) over D.

Much of the work that follows is devoted to explaining the general framework forcarrying out this type of construction and establishing its functoriality. We will seethat the essential property underlying the uniqueness of the preceding constructionis the unipotence of the π1-action.

2. de Rham theory for connections

Just as we uniquely extend the universal OY -linear derivation dX/Y : OX →Ω1

X/Y to a complex Ω•X/Y on the exterior powers of Ω1X/Y such that the connecting

maps satisfy the Leibnitz rule, we have a version for connections; one new aspectis that it is a non-trivial constraint that we get a complex:

CLASSICAL MOTIVATION FOR THE RIEMANN–HILBERT CORRESPONDENCE 7

Theorem 2.1. Let (E ,∇) be a coherent sheaf on X with a connection ∇ relativeto Y . There is a unique abelian-sheaf map ∇p : Ωp

X/Y ⊗ E → Ωp+1X/Y ⊗ E satisfying

∇p(ω ⊗ s) = dω ⊗ s + (−1)pω ∧∇(s) for each p ≥ 0 (so ∇0 = ∇), and we have

∇p+q((ωp ∧ ωq)⊗ s) = ∇p(ωp ⊗ s) ∧ ωq + (−1)pωp ∧∇q(ωq ⊗ s)

for local sections ωp of ΩpX/Y and ωq of Ωq

X/Y .The composites ∇p+1 ∇p vanish for all p if and only if ∇1 ∇0 = 0, and when

X is Y -smooth then this is equivalent to the general identity ∇[~v,~w] = [∇~v,∇~w] forlocal vector fields ~v and ~w on X relative to Y .

Strictly speaking, ∇p(ωp ⊗ s) ∧ ωq is to be computed using the natural localmapping Ωp+1

X/Y ⊗ E → Ωp+q+1X/Y ⊗ E defined by (·) ∧ ωq : Ωp+1

X/Y → Ωp+q+1X/Y .

Proof. See [4, pp. 9–11] for a proof when Y = SpC; that proof works verbatimin the relative case, and the hypothesis there that the coherent sheaves are vectorbundles is not used in the proofs. As with the construction of the de Rham complex,the essential point is existence of the ∇p’s; uniqueness is obvious.

The relevance of Y -smoothness in the final part of the theorem is that in thiscase we may establish identities among 1-forms by evaluation against relative vectorfields since Ω1

X/Y is its own double-dual.

The connection ∇ is integrable when ∇1 ∇0 = 0, and the resulting complex(Ω•X/Y ⊗OX

E ,∇•) is the de Rham complex of (E ,∇).

Example 2.2. In Example 1.1, the associated de Rham complex is obtained byapplying the functor Λ ⊗π−1OY

(·) to the usual OY -linear holomorphic de Rhamcomplex for X over Y . Thus, the connections in Example 1.1 are integrable. Inte-grability is also automatic when X is Y -smooth with fibers of dimension at most1, as then Ω2

X/Y = 0.

It is straightforward to check that formation of the de Rham complex of (E ,∇)is compatible with pullback, and that the operations of tensor product, duality, andpullback preserve integrability of connections.

Remark 2.3. An alternative description of the integrability condition may begiven in the spirit of Remark 1.3, as follows. Identify ∇ with a linear isomorphismξ : q∗2E ' q∗1E on the first infinitesimal neighborhood of the diagonal in X×Y X. Letqij be the projections from the first infinitesimal neighborhood of the triple-diagonalin X×Y X×Y X to the first infinitesimal neighborhood of the diagonal in X×Y X.Pulling back ξ along the qij ’s defines linear isomorphisms ξij : q∗j E ' q∗i E . It isnatural to ask if the cocycle condition ξ13 = ξ12 ξ23 holds. From the viewpoint ofuniversality, as in Remark 1.3, if we consider ξ as the specification of isomorphismsξq′1,q′2

: q′2∗F ' q′1

∗F for all pairs of Y -retractions q′1, q′2 : X ′ → X of a common

square-zero closed Y -immersion δ : X → X ′, then the cocycle condition is thetransitivity condition

ξq′1,q′3= ξq′1,q′2

ξq′2,q′3

for any triple of retractions of a common δ.It can be proved that when the cocycle condition holds, then ∇ must be inte-

grable, and that if X is Y -smooth then integrability forces the cocycle condition tohold; thus, an integrable connection on coherent sheaf on a smooth Y -space maybe viewed as first-order descent data. We will not make use of the equivalence of

8 BRIAN CONRAD

integrability and the cocycle condition in the smooth case, and so we leave it to theinterested reader to either carry out the verification or to consult [2, §2] for generalarguments presented in the Q-scheme setting (or [1, Ch. II, Prop. 3.3.9, Thm. 4.2.11]for proofs in the most general axiomatic framework).

Example 2.4. Assume that X is Y -smooth and that E is a vector bundle on X.Choose local relative coordinates z1, . . . , zn on an open U in X, with U sufficientlysmall so that there exists a basis ej of E |U . A connection ∇ on E relative to Yis determined by the Christoffel-symbol expansions

∇∂zi(ej) =

∑k

Γkijek.

We shall now formulate integrability of ∇|U as a system of differential equations inthe Christoffel symbols.

Since X is Y -smooth, by Theorem 2.1 the integrability of ∇|U is equivalent tothe vanishing of

(2.1) [∇~v,∇~w]−∇[~v,~w]

for local vector fields ~v and ~w over opens in U , and the Leibnitz rule implies that itsuffices to check such vanishing on a local basis of vector fields. Thus, integrabilityof ∇|U amounts to vanishing of (2.1) when ~v = ∂zi

and ~w = ∂zjfor all i, j. Let

∇i = ∇∂zi. Since [∂zi

, ∂zj] = 0, we need to compute ∇i ∇j −∇j ∇i.

Evaluating on ek gives

∇i(∇j(ek))−∇j(∇i(ek)) = ∇i

(∑s

Γsjkes

)−∇j

(∑s

Γsikes

)

=∑

`

(∑s

(ΓsjkΓ`

is − ΓsikΓ`

js) +

(∂Γ`

jk

∂zi− ∂Γ`

ik

∂zj

))e`.

If we let R`ijk denote the coefficient of e` in this sum, then the integrability of ∇|U

is equivalent to the system of differential equations R`ijk = 0. The reader may also

check (or see [4, p. 11]) that

(∇1 ∇0)(ek) =∑

`

∑i<j

R`ijk dzi ∧ dzj

⊗ e`,

so these R`ijk’s describe the operator ∇1 ∇0.

These considerations carry over to the case of C∞-manifolds, and in the specialcase when E = TX/Y is the tangent bundle, ei = ∂zi

, and ∇ is the Levi-Civitaconnection associated to a Riemannian metric, the R`

ijk’s are the coefficients of thecurvature tensor on a Riemannian manifold. In general, the operator ∇1 ∇0 iscalled the curvature of the connection, since we have seen above that it encodes theobstruction to finding local coordinates zi whose associated directional derivatives∇∂zi

commute with each other. (Recall that in multivariable calculus, the vanishingof d2 is equivalent to the identity ∂xi∂xj = ∂xj ∂xi .)

Example 2.5. Continuing the preceding example, let us give geometric formu-lations of the differential equation ∇(s) = 0 (for s ∈ E (U)) and the system ofdifferential equations R`

ijk = 0 that encode integrability of ∇|U . Let V = VE → X

be the geometric vector bundle with sheaf of sections E . Let vj be the linear

CLASSICAL MOTIVATION FOR THE RIEMANN–HILBERT CORRESPONDENCE 9

coordinates on V |U over U that are dual to the chosen basis ej of E |U . Thus,V |U is coordinatized over Y by the zi’s and vj ’s, and so the relative tangent bundleTV/Y is free over V |U on the basis of ∂zi ’s and ∂vj ’s. We may define vector fields

Xi = ∂zi−∑

k

∑j

vjΓkij(z)

∂vk

on V |U , and a direct calculation shows that

[Xi, Xj ] = −∑k,`

R`ijkvk∂v`

.

Thus, the vanishing of [Xi, Xj ] on V |U is exactly the integrability condition.The only intervention of the ∂zj ’s in Xi is through the term ∂zi , so the Xi’s

freely generate a vector bundle W over V |U that is a subbundle of the part of TV/Y

that lies over V |U , and this subbundle over V |U has rank equal to the (constant)relative dimension of X over Y . The absence of ∂zk

’s in the formula for [Xi, Xj ]implies that [Xi, Xj ] lies in W if and only if [Xi, Xj ] = 0. That is, integrabilityof ∇|U is equivalent to the property that the subbundle W of TV/Y |U over V |U isstable under the bracket on vector fields.

Let s : U → V |U be a section in E (U). The induced map dX/Y s : TX/Y |U →TV/Y |U is characterized by

∂zi7→ ∂zi

+∑

k

∂sk

∂zi∂vk

where sk = vk s, and an inspection of the definition of the Xj ’s shows that theonly possible O-linear relation of the form ds(∂zi

) =∑

j aijXj with aij ∈ O(V |U )is aij = 0 for i 6= j and aii = 1. Thus, ds factors through the subbundle W freelyspanned by the Xi’s if and only if ds(∂zi

) = Xi; i.e., if and only if

∂zi(sk) = −

∑j

sjΓkij

for all i and k. By Example 1.2, this final system of equations is exactly thecondition ∇(s) = 0.

We draw two geometric conclusions in terms of the vector bundle W : a sections of E over U is killed by ∇ if and only if its differential TX/Y |U → TV/Y |U factorsthrough W , and the integrability of ∇|U is exactly the condition that W is stableunder the bracket on TV/Y |U .

The importance of integrability is due to:

Theorem 2.6 (Riemann–Hilbert correspondence). Assume X is smooth, and let(E ,∇) be a vector bundle on X with an integrable connection. The sheaf ker∇is locally constant with C-rank equal to the O-rank of E , and the natural mapOX ⊗C ker∇ → E carrying ∇ back to dX ⊗ 1 is an isomorphism.

The functors (E ,∇) ker∇ and Λ (OX⊗CΛ,dX⊗1) are inverse equivalencesof categories between the category of vector bundles on X equipped with an integrableconnection and the category of local systems of finite-dimensional C-vector spaceson X. In particular, the category of such pairs (E ,∇) is abelian, with kernels andcokernels formed in the evident manner.

10 BRIAN CONRAD

Combining Example 1.1 and the Riemann–Hilbert correspondence, we see thereason for the terminology “integrability”: it is exactly the condition that ensureswe can locally (uniquely) solve the differential equation ∇(s) = 0 near any x0 ∈ Xfor an arbitrary initial condition s(x0) ∈ E (x0). Strictly speaking, the phrase“Riemann–Hilbert correspondence” usually refers to a vast generalization with per-verse sheaves replacing local systems and regular holonomic D-modules replacingvector bundles with integrable connection.

Proof. Once it is proved that ker∇ is a local system and that α : OX⊗Cker∇ → Eis always an isomorphism, the abelian-category claim follows from the fact that localsystems of finite-dimensional C-vector spaces on X form an abelian category in theevident manner. By Lemma 1.6, ker∇ is locally constant if its fiber-ranks arelocally constant, and for all x ∈ X the map (ker∇)x → E (x) is injective. Thus,since a map between vector bundles is an isomorphism if and only if it induces anisomorphism on fibers, it suffices to show that for any x0 ∈ X and any s0 ∈ E (x0),there exists a section s of E near x with s(x0) = s0 and ∇(s) = 0. This problem islocal near x0, so we may put ourselves in the local situation considered in Example2.5 with the base Y equal to SpC. Using notation as is introduced there, thecondition ∇(s) = 0 says that s : U → VE |U has differential factoring through thesubbundle W in the tangent bundle of VE |U ; moreover, the integrability of ∇|Uimplies that W is stable under the bracket.

The geometric form of the Frobenius integrability theorem [6, 1.64] says that asubbundle B of the tangent bundle of a C∞ (or real-analytic) manifold M is stableunder the bracket if and only if for each m ∈ M there is an open neighborhood Um

of m and a closed connected C∞-submanifold (or real-analytic submanifold) N inUm passing through m such that the tangent space Tn(N) ⊆ Tn(M) at each point nis B(n); the manifold N is called an integral manifold for the subbundle B, and it isuniquely determined in Um. This theorem carries over to the complex-analytic caseon complex manifolds because a closed C∞-submanifold N in a complex manifoldM is a complex submanifold if and only if Tn(N) is a C-linear subspace of Tn(M)for all n ∈ N (proof: if z1, . . . , zs are local coordinates on U ⊆ M containing n0 ∈ Nsuch that ∂z1 |n0 , . . . , ∂zr |n0 is a basis of the C-subspace Tn0(N) ⊆ Tn0(M), thenπ : U → Cr is a submersion at n0 with restriction πN : U ∩N → Cr that is a C∞

open immersion after shrinking U . The composite j of π−1N and the inclusion of

U∩N into M is a closed C∞-embedding of πN (U∩N) into U with image U∩N ; theinjective differential of j at each point of πN (U ∩N) is C-linear at each point sinceTn(N) is a C-linear subspace of Tn(M) for all n ∈ N , and so j is holomorphic).

Choose an initial condition s0 =∑

ajej(x0) ∈ E (x0), and consider this as apoint in VE |U over x0 ∈ U . Let W be the integral manifold for W through s0 in asmall open neighborhood of s0, so Tw(W ) = W (w) for all w ∈ W . The differentialof the projection W → U at s0 sends the basis Xi(s0) of W (s0) = Ts0(W ) tothe basis ∂zi

(x0) of Tx0(U), and so this projection is a local isomorphism. Itsanalytic inverse near x0 provides a local section σ : U → W satisfying σ(x0) = s0

after we shrink U around x0, and the composite s : U → W → VE |U has differentialfactoring through W since Tw(W ) = W (w) for all w ∈ W . Thus, s ∈ E (U) is inker∇ and satisfies the initial condition s(x0) = s0.

If (E ,∇) is a vector bundle with connection on a manifold, the flat sections(or horizontal sections) of E are the sections of the local system ker∇ ⊆ E . The

CLASSICAL MOTIVATION FOR THE RIEMANN–HILBERT CORRESPONDENCE 11

Riemann–Hilbert correspondence says that a vector bundle with integrable connec-tion may be reconstructed from its flat sections. This correspondence is very useful,since local systems are topological and vector bundles with connection require onlythe language of coherent sheaves (and so are algebraic).

Combining the Riemann–Hilbert correspondence with Example 2.2 and the holo-morphic Poincare lemma, we get:

Corollary 2.7 (Poincare lemma for integrable connections). If (E ,∇) is a vectorbundle with integrable connection on a manifold X, then the de Rham complex(Ω•X ⊗ E ,∇•) is a resolution of the local system ker∇.

It is a remarkable fact that if E is a coherent sheaf on a manifold X and E admitsan integrable connection, then E must be a vector bundle; we will not use this fact.A relativization of (our version of) the Riemann–Hilbert correspondence can alsobe proved: if π : X → Y is smooth and E is a coherent sheaf on X admitting anintegrable connection ∇ relative to Y , then ker∇ is a locally free π−1OY -moduleand OX ⊗π−1OY

ker∇ → E is an isomorphism. In particular, E is a vector bundleand the category of coherent sheaves on X with integrable connection relative toY is equivalent to the category of locally free π−1OY -modules. See [4, 2.23] for anelegant inductive proof of these facts.

3. Regular singularities in dimension 1

A very interesting feature of connections is that they can be used to canonicallyextend a vector bundle across a missing point in the 1-dimensional case. We sawa concrete example worked out at the end of §1, and we wish to study the generalproblem on an arbitrary complex manifold with pure dimension 1. Since integra-bility is automatic in the 1-dimensional case, we will not make explicit mention ofthis condition again; the higher-dimensional and relative generalizations of whatwe are about to do (see [4]) require integrability.

Definition 3.1. Assume X is smooth with pure dimension 1, and let X∗ be thecomplement of a 0-dimensional analytic set Σ in X, with j : X∗ → X the inclusion.Let (E ∗,∇∗) be a vector bundle with connection on X∗. This pair has regularsingularities along Σ if there exists a vector bundle E ⊆ j∗E ∗ on X such thatj∗(∇∗) carries E into Ω1(Σ)⊗ E .

In the setting of the definition, the induced map ∇ : E → Ω1(Σ) ⊗ E certainlysatisfies the Leibnitz rule, and the pair (E ,∇) is also called a vector bundle withconnection having regular singularities along Σ. The definition simply says thatE ∗ extends to a vector bundle E on X such that ∇∗ extends to a “meromorphicconnection” with at worst simple poles along Σ. Concretely, the condition on theconnection over X∗ is that for each any σ ∈ Σ there exists a basis of E ∗ over apunctured open neighborhood of σ such that the connection matrix has entries thatare meromorphic at σ with at worst a simple pole.

Since pullback along an analytic map of Riemann surfaces carries 1-forms withsimple poles to 1-forms with simple poles (proof: d(te)/te = e · dt/t), we see thatfor any analytic map (X ′,Σ′) → (X, Σ) we may define pullback for pairs (E ,∇) onX with regular singularities along Σ. Duality and tensor product are defined in theevident manner.

A natural question is whether any pair (E ∗,∇∗) on X∗ admits an extension to Xwith regular singularities along Σ, and whether such an extension is unique. We met

CLASSICAL MOTIVATION FOR THE RIEMANN–HILBERT CORRESPONDENCE 13

value v0 = q−N0 (v) at q0. Analytic continuation of this solution around an i-

counterclockwise generating loop though q0 gives the value e−2πiN (v0) = Ti(v0).Since we can find v such that v0 = q−N

0 (v) is any desired element of V , the i-counterclockwise generating loop of π1 acts as Ti on V . This settles the existenceproblem.

Meromorphicity. Working locally near the origin of a disc and trivializingE allows us to suppose E = OD ⊗C V for a finite-dimensional C-vector space V .Using the basis dq/q of Ω1

D(0), there is a unique

N ∈ Γ(D,E nd(E )) = HomC(V ,E ) = HomC(V,E (D))

such that ∇(v) = N(v)dq/q for local sections v of V ; in contrast with the existencestep, N might not be constant. We have to prove that if s ∈ E (D∗) satisfies∇(s) = 0, then s is meromorphic at the origin. We may identify N and s withholomorphic maps N : D → End(V ) and s : D∗ → V , and (as in Example 1.2) thecondition ∇(s) = 0 means s′(q) = −(N(q)(s(q)))/q on D∗.

To prove the meromorphicity of any such s, it will be convenient to use a Hermit-ian inner product H on V . Let us briefly recall some constructions with Hermitianinner products. The dual space V ∨ acquires an inner product H∨ by the require-ment H∨(H(·, v),H(·, v′)) = H(v′, v); i.e., the dual-basis to an orthonormal basis isorthonormal. Also, End(V ) = V ⊗V ∨ acquires a Hermitian inner product H ⊗H∨

by the requirement

(H ⊗H∨)(v ⊗ `, v′ ⊗ `′) = H(v, v′)H∨(`, `′);

explicitly, (H⊗H∨)(T, T ′) =∑

j H(T (ej), T ′(ej)) for any orthonormal basis ej ofV . Let || · || denote the associated norms on V and End(V ); it is clear that ||T (v)|| ≤||T || · ||v||, since any nonzero v may be scaled to become part of an orthonormal basis.

Choose any 0 < ε < 1 and define

C = sup|q|≤ε

||N(q)||, c = sup|q|=ε

||s(q)||.

For 0 < |q| ≤ ε, we shall prove that

(3.1) ||s(q)|| ≤ c

(|q|ε

)−C

,

so s is meromorphic at the origin with ord0(s) ≥ −C.Pick q0 with |q0| = ε, and consider the function σ(r) = ||s(rq0)||2 for r ∈ (0, 1].

We will prove σ(r) ≤ σ(1)r−2C . Since σ(1) ≤ c2, extracting square roots andsetting q = rq0 will then give the desired result (3.1) as we vary r over (0, 1] andvary q0 over the circle of radius ε. Using the identity σ(r) = H(s(rq0), s(rq0)),clearly

σ′(r) = H(∂r(s(rq0)), s(rq0)) + H(s(rq0), ∂r(s(rq0))),where ∂r = d/dr. It is also clear that ∂r(s(rq0)) = q0s

′(rq0), so

||∂r(s(rq0))|| = ε||s′(rq0)||.By the Cauchy–Schwarz inequality |H(v, v′)| ≤ ||v|| · ||v′|| we obtain

|σ′(r)| ≤ 2||s(rq0)|| · ||∂r(s(rq0))|| = 2ε||s(rq0)|| · ||s′(rq0)||.The equation s′(q) = −(N(q)(s(q)))/q implies ||s′(q)|| ≤ C||s(q)||/|q|, so

−σ′(r) ≤ |σ′(r)| ≤ 2εC

|rq0|· ||s(rq0)||2 =

2C

r· σ(r).

14 BRIAN CONRAD

We claim that σ has no zeros on (0, 1] unless it is identically zero. Supposeσ(r0) = 0 for some r0 ∈ (0, 1]. We may write σ(r) = (r − r0)nh(r) with n > 0 anda real-analytic h (if r0 = 1, note that σ extends real-analytically past 1), so

2C

r· |r − r0|n|h(r)| = 2C

r· |σ(r)| ≥ |σ′(r)| = |r − r0|n−1|nh(r) + (r − r0)h′(r)|.

Cancelling |r − r0|n−1 and evaluating at r0 gives h(r0) = 0. Thus, if σ vanishes atsome point in (0, 1] then it has infinite order of vanishing there, and so σ = 0 byreal-analyticity.

Since (3.1) for q = rq0 is obvious if σ = 0, it remains to consider the case whenσ is everywhere positive on (0, 1]. In this case,

(log σ)′ =σ′

σ≥ −2C

r= −(log(r2C))′.

Integration over [r, 1] yields

log(

σ(r)σ(1)

)≥ log(r−2C).

Exponentiation gives the desired estimate.

Definition 3.3. Let OXΣ ⊆ j∗OX∗ be the subsheaf of sections that are meromor-phic along Σ. A meromorphic vector bundle on X along Σ is a locally free sheaf offinite rank over OXΣ .

The functor OXΣ ⊗OX(·) carries holomorphic vector bundles on X to meromor-

phic vector bundles on X along Σ, compatibly with tensor products and duality andpullback, and clearly this functor is faithful and essentially surjective. In particular,Ω1

X gives rise to a meromorphic vector bundle along Σ that we denote Ω1XΣ

. We canuse the Leibnitz rule to define the concept of a connection ∇ : E → Ω1

XΣ⊗ E on a

meromorphic vector bundle on X along Σ. Explicitly, such connections arise frommaps ∇ : E → Ω1

X(D) ⊗ E satisfying the Leibnitz rule, where E is a holomorphicvector bundle on X and D is an effective divisor supported in Σ.

Let MCXΣ be the category whose objects are pairs (E ,∇) consisting of a holo-morphic vector bundle on X endowed with a connection ∇ having regular singu-larities along Σ, and whose morphisms are maps of meromorphic vector bundles onX along Σ that are compatible with the associated meromorphic connections; notethat such a map is automatically holomorphic on the vector bundles over X∗ be-cause these bundles are O-free over the locally constant C-module of flat sections.We may therefore rephrase Theorem 3.2 as saying that there is an equivalence ofcategories between local systems on X∗ and MCXΣ .

We would like to refine this equivalence by singling out a special class of regularsingular connections that are functorially determined by their local systems of flatsections on X∗ without the interference of meromorphic vector-bundle maps overX. This will clarify the uniqueness of the construction at the end of §1.

4. Residues, monodromy, and canonical extensions

In the proof of the existence of regular-singular extensions, there was some free-dom of choice of N : we only required e−2πiN = Ti. If N = Nss + Nn is the addi-tive Jordan decomposition of N as a sum of commuting semisimple and nilpotentendomorphisms, then e−2πiNsse−2πiNn is the multiplicative Jordan decompositionTi,ssTi,u of Ti. In particular, the nilpotent Nn is uniquely determined by the local

CLASSICAL MOTIVATION FOR THE RIEMANN–HILBERT CORRESPONDENCE 15

monodromy action Ti: it is − log(Ti,u)/2πi (this is independent of i). However,Nss is not unique: the only requirement is that it preserves each eigenspace ofTi,ss and acts semisimply on the λ-eigenspace of Ti,ss with eigenvalues µk satisfyinge−2πiµk = λ. There is Z-ambiguity in each µk. One way to eliminate this ambi-guity is to require N to be nilpotent, but this can only be done when the localmonodromy is unipotent.

In order to better understand the meaning of N and its relationship with Ti, it isconvenient to introduce some more terminology. Fix a basis 2πi of Z(1), and choosea small open disc D ⊆ X centered at σ ∈ Σ and not meeting any other point of Σ.Let D∗ = D−σ. The isomorphism OX∗ ⊗C Λ∗ ' E ∗ identifies Λ∗x and E ∗(x) forx ∈ X∗, and so for x ∈ D∗ we get the monodromy transformation Tx,i ∈ Aut(E ∗(x))at x; concretely, for v ∈ E ∗(x), the differential equation ∇∗(s) = 0 with initialcondition s(x) = v may be locally uniquely solved along an i-counterclockwisegenerating loop of π1(D∗, x), and this solution specializes to Tx,i(v) after goingonce around the loop. The map x 7→ Tx,i is an analytic section of E nd(E ∗) over D∗

since a solution to a first-order vector-valued ODE has analytic dependence on bothinitial values and auxiliary parameters (this fact follows from the local constructionof solutions via uniform limits of Picard-iterates).

The equivalence between local systems and representations of the fundamen-tal group implies that the monodromy transformation Tx,i at one point x ∈ D∗

uniquely determines (E ∗,∇∗)|D∗ , but only up to non-unique isomorphism since thepair (E ∗x , Tx,i) admits non-trivial automorphisms. Explicitly, we use Tx,i to recon-struct the local system Λ∗|D∗ (up to non-unique isomorphism) as a representationof π1(D∗, x), where the i-counterclockwise generator acts as Tx,i on the x-fiber, andthen (E ∗,∇∗)|D∗ is associated to this local system as in Example 1.1.

A pair (E ∗,∇∗) has unipotent monodromy relative to Σ when the local mon-odromy representations Tx,i near each σ ∈ Σ are unipotent automorphisms; i.e.,the fundamental group of a small punctured disc at σ acts by unipotent automor-phisms on the fiber of the space of flat sections at x near σ. Let (E ,∇) be aholomorphic vector bundle on X endowed with a connection having regular sin-gularities along Σ, with (E ∗,∇∗) its restriction to X∗; we have seen that such an(E ,∇) may always be found when (E ∗,∇∗) is given. Our goal is to single out apreferred (E ,∇) when (E ∗,∇∗) is given; the criterion for the existence of a pre-ferred extension across X will be given in terms of a differential-equations analogueof the local monodromy transformation, called the residue of the regular-singularconnection at points of Σ.

The starting point for the construction of the residue is the observation that forlocal sections s of E and f of OX near σ ∈ Σ such that f(σ) = 0,

∇(fs) = df ⊗ s + f · ∇(s)

is a local section of Ω1X(Σ)⊗E near σ whose image in the fiber at σ is killed by the

map

Ω1X(Σ)σ ⊗C E (σ) resσ⊗1→ E (σ)

since the 1-form df is holomorphic near σ and f(σ) = 0 (thereby killing any simplepole of ∇(s) at σ). Thus, even though ∇ is not OX -linear, the composite of itsinduced map on σ-stalks and the residue-map at σ gives a well-defined C-linearmap of fibers

Resσ(∇) : E (σ) → E (σ);

16 BRIAN CONRAD

this C-linear endomorphism of E (σ) is the residue of ∇ at σ.To make residues explicit, choose a local trivialization E |D ' OD ⊗C V over an

open disc D around σ with coordinate q, with V a finite-dimensional C-vector space.We have a holomorphic map N : D∗ → End(V ) characterized by the condition

∇(fv)(q) = (df)(q)⊗ v +dq

q⊗ f(q) ·N(q)(v)

for v ∈ V and a section f of OD. The residue Resσ(∇) ∈ End(V ) = End(E (σ)) isN(0). The connection matrix Γ = N/q depends on the choice of local trivializationof E and local coordinate q, but its residue N(0) as an element of End(E (σ)) isindependent of these choices because we gave an intrinsic construction above.

The relationship between the local residue of a regular-singularities connectionand the local monodromy transformation of the associated local system of flatsections is that the residue is a limiting logarithm of the monodromy:

Theorem 4.1. Fix a local trivialization E |D ' OD ⊗C V for a finite-dimensionalC-vector space V . Using the resulting isomorphisms E (q) ' V for q ∈ D, theanalytic map D∗ → End(V ) defined by q 7→ Tq,i ∈ End(E (q)) = End(V ) extendsanalytically across the origin with value e−2πiRes0(∇) ∈ End(E (0)) = End(V ) at theorigin.

In particular, the characteristic polynomial of e−2πiRes0(∇) is equal to the com-mon characteristic poynomial of the monodromy transformations Tx,i for x ∈ D∗,and so the monodromy is unipotent if the residue of the connection has all of itseigenvalues in Z.

Proof. The problem is local; let us formulate it in concrete terms. Choose ananalytic map N : D → End(V ), and for q0 ∈ D∗ consider the V -valued differentialequation

s′(q) = −N(q)(s(q))q

on D∗ with initial condition s(q0) = v. Let hi ⊆ C−R be the connected componentof our choice of basis 2πi for Z(1). Pulling back to hi via z 7→ e2πiz gives the V -valued differential equation f ′(z) = −2πi·N(e2πiz)(f(z)) on hi with initial conditionf(z0) = v for a choice of z0 such that e2πiz0 = q0. The unique solution has valueTq0,i(v) ∈ V at z0 + 1; this is independent of the choice of z0 over q0. We want tounderstand the behavior of Tq,i ∈ End(V ) for q near 0.

Since N(q) = N(0) + qN1(q) with N(0) ∈ End(V ) and N1 : D → End(V )holomorphic, the differential equation for f on hi may be written

(4.1) g′y(x) = −2πi(N(0) + e2πizN1(e2πiz))(gy(x))

where gy(x) = f(x + iy) ∈ V . The shift z 7→ z + 1 leaves the differential equationinvariant, and for z = x + iy ∈ hi we want to study the operator Tz : V → Vsending v ∈ V to the value at x + 1 of the solution gy whose value at x is v. Since(4.1) encodes analytic continuation of a flat section to an integrable connection(on a punctured disc), and the sheaf of flat sections is locally constant (by theRiemann–Hilbert correspondence), it follows that gy is either nowhere-vanishing oris identically zero. This strong property of the gy’s ultimately rests on the Frobeniusintegrability theorem.

Our interest is in the behavior of Tz(v) as y → ∞ and x remains in a boundedinterval. More precisely, for the unique solutions gy to (4.1) (with z = x + iy)

CLASSICAL MOTIVATION FOR THE RIEMANN–HILBERT CORRESPONDENCE 17

on [−1, 1] such that gy(−1) ∈ V runs through a fixed basis B of V , we need tostudy the relationship between gy(x) and Tx+iy(gy(x)) = gy(x + 1) for x ∈ [−1, 0];note that as we vary the initial condition gy(−1) through B, the values gy(x) runthrough a basis of V for each x ∈ [−1, 0] (due to the linearity of (4.1) and thenowhere-vanishing property for its nonzero solutions that we noted above).

The differential equation for gy is g′y(x) = A(x, y)(gy(x)) where

A : (x, y) 7→ A(x, y) = −2πi(N(0) + e2πi(x+iy)N1(e2πi(x+iy))) ∈ End(V )

is real-analytic. Since N1(q) is bounded for q near zero, the exponential decay ofe−2πy implies that as y → ∞ and x remains bounded, A converges uniformly tothe constant map (x, y) 7→ −2πiN(0). It follows that as y → ∞, the solutiongy,b : [−1, 1] → V satisfying gy,b(−1) = b ∈ B converges uniformly to the unique gb

satisfying g′b(x) = −2πiN(0)(gb(x)) with gb(−1) = b.Since gy,b(x)b∈B is a basis of V uniformly converging to the basis gb(x)b∈B

of V as y → ∞ (with x ∈ [−1, 1]), it follows that as y → ∞ the operatorsTx+iy ∈ End(V ) converge uniformly in x ∈ [−1, 0] to the monodromy opera-tor Ti for the differential equation g′(x) = −2πiN(0)(g(x)). Hence, q 7→ Tq,i

has a removable singularity at q = 0 with value Ti at the origin. The equationg′(x) = −2πiN(0)(g(x)) can be explicitly solved: g(x) = e−2πixN(0)(v0) for v0 ∈ V .Thus, g(x + 1) = e−2πiN(0)(g(x)) for any solution g and any x, so

Ti = e−2πiN(0) = e−2πiRes0(∇) ∈ End(V ).

The relationship between residues and monodromy shows that if we are givena local system Λ∗ of finite-dimensional C-vector spaces on X∗, and (E ∗,∇∗) isthe corresponding vector bundle with connection on X∗, a regular-singularitiesextension (E ,∇) on X specifies a limiting logarithm of the local monodromy. Ingeneral there is no canonical logarithm for an automorphism of a vector space;however, a unipotent automorphism has a unique nilpotent logarithm via the serieslog(1 + (U − 1)) that is polynomial in U for unipotent U . Hence, it is reasonableto focus attention on the case when the local monodromy of Λ∗ at each σ ∈ Σ isunipotent; in this case, our proof of existence of a regular-singularities extensions(E ,∇) yields an extension with nilpotent residues along Σ by choosing the localconstant N to be the logarithm of the unipotent local monodromy. Fortunately,the (E ,∇) constructed in this way is functorial in Λ∗:

Theorem 4.2. Let VCnilX be the category of pairs (E ,∇) on X consisting of a

vector bundle E on X and a connection ∇ having regular singularities and nilpotentresidues along Σ. Let LSuni

X∗ be the category of local systems of finite-dimensionalC-vector spaces on X∗ with unipotent local monodromy.

The functor (E ,∇) ker∇|X∗ is an equivalence of categories from VCnilX to

LSuniX∗ , and flat sections in E (X∗) extend holomorphically to E (X).

In concrete terms, the final assertion in the theorem says that when the con-nection has nilpotent residues, any flat section on a punctured disc (centered at apoint of Σ) extends to a holomorphic section of the vector bundle.

Proof. We have already seen that the functor does carry VCnilX to LSuni

X∗ , and that itis faithful and essentially surjective. The Hom-trick used in the proof of Theorem

18 BRIAN CONRAD

3.2 reduces the full faithfulness to the holomorphicity on X for flat sections overX∗.

The problem is local, and so we are reduced to considering the following situation.Let N : D → End(V ) be holomorphic with N(0) nilpotent. We must prove thatif s : D∗ → V is holomorphic and s′(q) = −(N(q)(s(q)))/q, then s is holomorphicat the origin. Pick a Hermitian inner product on V . We will prove that for anysolution s on a sector U , there is a positive integer d such that ||s(q)|| = O((log |q|)d)as q → 0 in U ; in fact, d = dim V will suffice. This implies that if s is defined onD∗ then qs(q) → 0 as q → 0. Thus, s has a removable singularity at the origin, asdesired.

Let ∇ and ∇0 denote the regular-singular connections on E = OD ⊗ V given by

∇(f ⊗ v) = df ⊗ v + f · dq

q⊗N(q)(v), ∇0(f ⊗ v) = df ⊗ v + f · dq

q⊗N(0)(v).

Fix a sector U in D∗, and let e and e0 denote bases of the associated local systemsof flat sections on the simply connected manifold U . These give bases of E |U ,due to the equivalence between local systems and vector bundles with integrableconnection, and so there is a matrix M = (mij) of holomorphic functions on Usuch that ej =

∑i mije0,i for all j.

Since∇(s) = ∇0(s) + dq ⊗N1(q)(s)

for any section s of E , where N = N(0) + qN1 with N1 holomorphic, we have∇(e0,j) = dq ⊗ N1(q)(e0,j) for all j. Applying ∇ to the identity ej =

∑mije0,i

therefore gives

0 = ∇(ej) =∑

i

dmij ⊗ e0,i + dq ⊗∑

i

mijN1(q)(e0,i)

on U . Equivalently,m′

ij(q) = −∑

k

mkj(q)νik(q)

where N1(q)(e0,i(q)) =∑

νki(q)e0,k(q). That is, we have the matrix equation M ′ =[N1]e0 ·M where [N1]e0 has value at q ∈ U equal to the matrix of N1(q) ∈ End(V )with respect to the basis e0,i(q) of V .

Fix a basis v of V . For q ∈ U , let P (q) be the change-of-basis matrix thatconverts e0(q)-coordinates into v-coordinates, so the matrix

[N1]v = P [N1]e0P−1

for N1 with respect to the constant basis v of OD ⊗C V is holomorphic on D sinceN1 : D → End(V ) is analytic, and so this matrix is bounded near the origin. Ourdifferential matrix equation on U is

M ′ = P−1[N1]vPM.

The columns of P−1 express the v-coordinates of the e0,i’s, and so our generallocal solution q−N(0)(v0) in the constant-connection case with nilpotent N(0) showsthat each entry of P−1(q) is a polynomial in log q with degree ≤ d = dim V . Thus,the entries in P−1(q) are O((log |q|)d) as q → 0 in U . Cramer’s rule implies asimilar conclusion for P (q) with a larger universal exponent replacing d. Thus, fora fixed choice of 0 < ε < 1 and all q ∈ U with 0 < |q| ≤ ε ≤ 1 we have

(4.2) ||M ′(q)||std ≤ B(log(1/|q|))m||M(q)||std

CLASSICAL MOTIVATION FOR THE RIEMANN–HILBERT CORRESPONDENCE 19

for some B ≥ 0 and some positive integer m; we are using the standard norm ond× d matrices that arises from the standard norm on Cd.

We will now prove that this estimate on M ′ forces ||M(q)||std to be bounded on Unear the origin, and so the estimate ||e0,i(q)|| = O((log |q|)d) as q → 0 in U impliesthe same for the ||ej(q)||’s, completing the proof. Fix 0 < ε < 1 and let

c = sup|q|=ε,q∈U

||M(q)||std < ∞;

finiteness is due to the fact that the supremum is taken over a set of q’s in Ubounded away from the origin (non-compactness of q ∈ U | |q| = ε is harmlesshere). Choose q0 ∈ U satisfying |q0| = ε, and define σ(r) = ||M(rq0)||2std for positiver in an open interval containing (0, 1]. Our goal is to show that the real-analytic σon (0, 1] is bounded by a positive constant independent of q0.

Arguing exactly as in the proof of Theorem 3.2, but using the estimate (4.2) toreplace the role of the estimate ||s′(q)|| ≤ C||s(q)||/|q| used in the proof of Theorem3.2, we conclude that

−σ′(r) ≤ |σ′(r)| ≤ 2B(log(1/εr))mσ(r)

and that σ : (0, 1] → R is either identically zero or is everywhere positive. Thecase σ = 0 is trivial, so we may assume that σ is everywhere positive. Dividing theinequality by σ(r) and integrating gives the estimate

log(

σ(r)σ(1)

)≤ 2B

∫ 1

r

(log1εt

)m dt =2B

ε

∫ ε

εr

(log1y)m dy =

2B

ε

∫ 1/εr

1/ε

(log x)m

x2dx.

Taking r → 0+ makes the right side increase up to an improper integral that isis finite and independent of q0, so exponentiation gives σ(r) ≤ C ′σ(1) ≤ C ′c2 forsome positive constant C ′ that is independent of q0.

We have proved that if Λ∗ is a local system of finite-dimensional C-vector spaceson X∗, and Λ∗ has unipotent local monodromy along Σ, then there is a functoriallyassociated vector bundle E on X extending E ∗ = OX∗ ⊗C Λ∗. The bundle E is thecanonical extension (also called the regular-singularities extension) of E ∗, thoughof course it also depends on the specification of ∇∗ (as the pair (E ∗,∇∗) determinesΛ∗). This implies that the explicit construction with elliptic-curve homology at theend of §1 is not ad hoc. As a special case of our general conclusions, unipotentlocal monodromy along Σ is trivial if and only if the connection on the canonicalextension has vanishing residues; i.e., the local system Λ∗ extends (necessarilyuniquely and functorially) to a local system Λ on X if and only if the canonicalextension of OX∗ ⊗C Λ∗ has a holomorphic connection, and then this canonicalextension is OX ⊗C Λ equipped with its standard holomorphic connection dX ⊗ 1on X.

The canonical extension E may also be characterized by the property that flatsections of E and E ∨ on a sector have logarithmic growth near the origin. There is ahigher-dimensional version of all of these results, for local systems having unipotentlocal monodromy with respect to a normal crossings divisor. See [4, Ch. II, 5.2] forfurther details.

References

[1] P. Berthlot, Cohomologie cristalline des varietes de caracterisque p > 0, Lecture Notes in

Mathematics 407, Springer-Verlag, Berlin, 1974.

20 BRIAN CONRAD

[2] P. Berthelot and A. Ogus, Notes on crystalline cohomology, Mathematical Notes 21, P. U.

Press, Princeton, 1978.

[3] A. Borel, Linear algebraic groups (2nd ed.), Springer-Verlag, New York, 1991.[4] P. Deligne, Equations differentielles a points singuliers reguliers, Springer–Verlag, New York,

1970.

[5] J. Dieudonne, A. Grothendieck, Elements de geometrie algebrique, Publ. Math. IHES, 4, 8,11, 17, 20, 24, 28, 32, (1960-7).

[6] F. Warner, Foundations of differentiable manifolds and Lie groups, Springer-Verlag, New

York, 1983.

E-mail address: bdconrad@umich.edu